Math / Geometry

Question $\begin{array}{l} k= \end{array}$ Check

Studdy Solution

STEP 1

1. The angles $CFD$ and $EFD$ are adjacent and form a straight line, meaning they are supplementary.
2. The sum of supplementary angles is $180^\circ$ .

STEP 2

1. Set up the equation for supplementary angles.
2. Solve for $k$ .
3. Verify the solution.

STEP 3

Since angles $CFD$ and $EFD$ are supplementary, their measures add up to $180^\circ$ . Set up the equation:
$(-9k + 170) + (5k + 19) = 180$

STEP 4

Combine like terms in the equation:
$-9k + 5k + 170 + 19 = 180$

STEP 5

Simplify the equation:
$-4k + 189 = 180$

STEP 6

Isolate the variable $k$ by subtracting $189$ from both sides:
$-4k = 180 - 189$

STEP 7

Simplify the right side:
$-4k = -9$

STEP 8

Divide both sides by $-4$ to solve for $k$ :
$k = \frac{-9}{-4}$

STEP 9

Simplify the fraction:
$k = \frac{9}{4}$

STEP 10

Verify the solution by substituting $k = \frac{9}{4}$ back into the expressions for the angles and checking their sum:
Angle $CFD$ : $-9\left(\frac{9}{4}\right) + 170 = -\frac{81}{4} + 170$
Angle $EFD$ : $5\left(\frac{9}{4}\right) + 19 = \frac{45}{4} + 19$
Calculate the sum of the angles and verify it equals $180^\circ$ .
The value of $k$ is:
$\boxed{\frac{9}{4}}$

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